The dense model theorem

A key component in the work of Green, Tao, and Ziegler on arithmetic and polynomial progressions in the primes is the dense model theorem. Roughly speaking this theorem allows one to model a dense subset of a sparse pseudorandom set by dense subset of the the ambient space. In the work of Green, Tao, and Zeigler this enabled them to model (essentially) the characteristic function of the set of primes with (essentially) the characteristic function of a set of integers with greater density. They then were able to obtain the existence of certain structures in the model set via Szemerédi’s theorem and its generalizations.

More recently, simplified proofs of the dense model theorem have been obtained independently by Gowers and Reingold, Trevisan, Tulsiani and Vadhan. In addition, the latter group has found applications of these ideas in theoretical computer science. In this post we give an expository proof of the dense model theorem, substantially following the paper of Reingold, Trevisan, Tulsiani and Vadhan.

With the exception of the min-max theorem from game theory (which can be replaced by (or proved by) the Hahn-Banach theorem, as in Gowers’ approach) the presentation is self-contained.

(We note that the the theorem, as presented below, isn’t explictly stated in the Green-Tao paper. Roughly speaking, these ideas can be used to simplify/replace sections 7 and 8 of that paper.)

— The dense model theorem —

Let . We’ll primarily be interested in certain real-valued functions on the set . We define the expectation of a function, say , on to be and the inner product of to be . We will call a non-negative real-valued function, say , on a measure (This isn’t a measure in the analytic sense however this terminology has become standard in the literature on the subject) if .

We will call a function on , say , bounded if . Analogously a measure, say , will be called bounded if . For a fixed finite collection of bounded functions, , on , we say that two measure, say and , are -indistinguishable with respect to if for every . Furthermore, a measure on is said to be -pseudorandom with respect to if the measures and are -indistinguishable. Here denotes the characteristic function of the set .

In addition to the set we will also need to consider the larger class of functions . We can now state the dense model theorem.

Theorem 1 Fix and , a finite collection of bounded functions on . Furthermore, let be a -pseudorandom measure with respect to the set and a measure majorized by . There exists and (that depend\footnote{For the sake of simplicity we will not work out the dependency of these parameters on . We do however (very briefly) discuss the dependencies in the remark at the end of this section.} only on ) and a bounded measure such that and and are -indistinguishable with respect to .

The thrust of the theorem is that and depend only on and not . At first the fact that is used in the hypothesis of the theorem and in the conclusion may seem strange. In applications, however, one often wishes to find a dense (-indistinguishable) model for a measure for a prescribed . One proceeds by locating , a majorant of , that is -indistinguishable from the measure . With applications of this form in mind, the statement of the theorem may seem more natural.

We will split the proof of the theorem into several parts/lemmas. Throughout will denote the set of bounded measures of expectation . We’ll typically denote an element of with the symbol .

Lemma 2Let and denote the convex hull of . Furthermore let be a real-valued polynomial (depending only on ) that maps to . If there is a function of the form with that -distinguishes from then there exists a function that -distinguishes from .

Proof: We note that and are -distinguishable with respect to if and only if they are -distinguishable with respect to (this allows us to remove the absolute value from the definition of -distinguishability given above).

Next we note that it suffices to show that and are -distinguishable with respect to , the convex hull of . To see this assume that -distinguishes and with , , and . We then have that , which easily implies that for some .

Furthermore, let be a real-valued polynomial that depends only on . We claim that it then suffices to show that there exists a function such that -distinguishes and . To see this set equal to the magnitude of the largest coefficient of and the degree of . Letting we have that hence

and thus, for some ,

Since the right-hand side depends only on the proof is complete.

It now suffices to assume that the conclusion of the the theorem is false and find a function of the form with that -distinguishes from , which would provide a contradiction.

Lemma 3 Assume that for every there exists a such that (in other words, assume the theorem is false). Then there exists a function that -distinguishes every function from . This is to say

for every .

Proof: Let denote a finite set\footnote{To see that such a set exists consider .} of bounded measures on such that the convex hull of is . Consider the matrix with entries . By the min-max theorem there exists and such that for all and for every .

By the hypothesis of the theorem we have that for every there exists such that . Thus there exists a such that . Hence and taking completes the proof.

We now let be a set of elements of that maximizes the quantity . Additionally let denote an element of the set that maximizes . Define . By construction we have that thus and . This implies that -distinguishes from .

Lemma 4Let where . Then there exists a threshold such that .

Proof: We have previously observed that . Using the fact that we have that . Combining this with the observation that we have that

Assuming that the conclusion of the lemma is false, that is , we can conclude

This would, however, contradict the inequality

which is an easy consequence of the inequality derived in the second sentence of this proof. Hence the proof is complete.

Lemma 5We have that for every .

Proof: Recall that . If for any then must vanish identically on . However this implies that

Let us briefly summarize the strategy for completing the proof of the theorem. From the previous lemma we have . However, since distinguishes and it must also distinguish and . This would contradict the hypothesis of the theorem if . In light of Lemma 2 it then suffices to show that can be approximated by a function of the form where . For this purpose, let be a polynomial mapping into and satisfying for and for . (The existence of such a polynomial can be obtained from standard variants of Weierstrass’ approximation theorem.)

Lemma 6Let be as defined above. Then -distinguishes from .

Proof: From the definition of we have that

Using the lower bound in (1), the pointwise inequality , Lemma 4 and Lemma 5 (in this order) we have that

Recalling that we may apply Lemma 2 to complete the proof of the dense model theorem.

Remark 1 We note that the only step of the proof where we haven’t explicitly recorded the relationship between , and is in the dependency of the polynomial on . Using a quantitative form of Weierstrass’ approximation theorem one can obtain and . We refer the reader here for details.